Utility

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Utility
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UTILITY FUNCTIONS

• A preference relation that is complete,
  reflexive, transitive and continuous can be
  represented by a continuous utility function (as
  an alternative, or as a complement, to the
  indifference map of the previous lecture).
• Continuity means that small changes to a
  consumption bundle cause only small changes to
  the preference (utility) level.

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UTILITY FUNCTIONS

• A utility function U(x) represents a preference
  relation if and only if x0 x1
  U(x0) gt U(x1) x0 x1
  U(x0) lt U(x1) x0 x1
  U(x0) U(x1)

p
p
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UTILITY FUNCTIONS

• Utility is an ordinal (i.e. ordering or ranking)
  concept.
• For example, if U(x) 6 and U(y) 2 then bundle
  x is strictly preferred to bundle y. However, x
  is not necessarily three times better than y.

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UTILITY FUNCTIONSand INDIFFERENCE CURVES

• Consider the bundles (4,1), (2,3) and (2,2).
• Suppose (2,3) gt (4,1) (2,2).
• Assign to these bundles any numbers that preserve
  the preference orderinge.g. U(2,3) 6 gt
  U(4,1) U(2,2) 4.
• Call these numbers utility levels.

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UTILITY FUNCTIONSand INDIFFERENCE CURVES

• An indifference curve contains equally preferred
  bundles.
• Equal preference ? same utility level.
• Therefore, all bundles on an indifference curve
  have the same utility level.

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UTILITY FUNCTIONSand INDIFFERENCE CURVES

• So the bundles (4,1) and (2,2) are on the
  indifference curve with utility level U º 4
• But the bundle (2,3) is on the indifference curve
  with utility level U º 6

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UTILITY FUNCTIONSand INDIFFERENCE CURVES
x2
(2,3) gt (2,2) (4,1)
x2
U º 6
U º 4
x1
xx11
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UTILITY FUNCTIONSand INDIFFERENCE CURVES

• Comparing more bundles will create a larger
  collection of all indifference curves and a
  better description of the consumers preferences.

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UTILITY FUNCTIONSand INDIFFERENCE CURVES
x2x2
U º 6
U º 4
U º 2
xx1 1
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UTILITY FUNCTIONSand INDIFFERENCE CURVES

• Comparing all possible consumption bundles gives
  the complete collection of the consumers
  indifference curves, each with its assigned
  utility level.
• This complete collection of indifference curves
  completely represents the consumers preferences.

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UTILITY FUNCTIONSand INDIFFERENCE CURVES

• The collection of all indifference curves for a
  given preference relation is an indifference map.
• An indifference map is equivalent to a utility
  function each is the other.

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UTILITY FUNCTIONS

• If
• (i) U is a utility function that represents a
  preference relation and (ii) f is a strictly
  increasing function,
• then
• V f(U) is also a utility functionrepresenting
  the original preference function.
• Example? V 2.U

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GOODS, BADS and NEUTRALS

• A good is a commodity unit which increases
  utility (gives a more preferred bundle).
• A bad is a commodity unit which decreases utility
  (gives a less preferred bundle).
• A neutral is a commodity unit which does not
  change utility (gives an equally preferred
  bundle).

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GOODS, BADS and NEUTRALS
Utility
Utilityfunction
Units ofwater aregoods
Units ofwater arebads
Water
x
Around x units, a little extra water is a
neutral.
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UTILITY FUNCTIONS
Cobb-Douglas Utility Function
Perfect Substitutes Utility Function
Note MRS (-)a/b
Perfect Complements Utility Function
Note MRS ?
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UTILITY

• Preferences can be represented by a utility
  function if the functional form has certain
  nice properties
• Example Consider U(x1,x2) x1.x2
• ?u/?x1gt0 and ?u/?x2gt0
• Along a particular indifference curve
• x1.x2 constant ? x2c/x1
• As x1 ? ? x2?
• i.e. downward sloping indifference curve

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UTILITY

• Example U(x1,x2) x1.x2 16
• X1 X2 MRS
• 1 16
• 2 8 (-) 8
• 3 5.3 (-) 2.7
• 4 4 (-) 1.3
• 5 3.2 (-) 0.8
• 3. As X1 ? MRS ? (in absolute terms), i.e
  convex preferences

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COBB DOUGLAS UTILITY FUNCTION

• Any utility function of the form
  U(x1,x2) x1a x2bwith a gt 0 and b gt 0 is
  called a Cobb-Douglas utility function.
• Examples
• U(x1,x2) x11/2 x21/2 (a b 1/2)V(x1,x2)
  x1 x23 (a 1, b 3)

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COBB DOUBLAS INDIFFERENCE CURVES
x2
All curves are hyperbolic,asymptoting to, but
nevertouching any axis.
x1
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PERFECT SUBSITITUTES

• Instead of U(x1,x2) x1x2 consider
  V(x1,x2) x1 x2.

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PERFECT SUBSITITUTES
x2
x1 x2 5
13
x1 x2 9
9
x1 x2 13
5
V(x1,x2) x1 x2
5
9
13
x1
All are linear and parallel.
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PERFECT COMPLEMENTS

• Instead of U(x1,x2) x1x2 or V(x1,x2) x1
  x2, consider W(x1,x2)
  minx1,x2.

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PERFECT COMPLEMENTS
x2
45o
W(x1,x2) minx1,x2
minx1,x2 8
8
minx1,x2 5
5
3
minx1,x2 3
3
5
8
x1
All are right-angled with vertices/corners on a
ray from the origin.
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MARGINAL UTILITY

• Marginal means incremental.
• The marginal utility of product i is the
  rate-of-change of total utility as the quantity
  of product i consumed changes by one unit i.e.
  

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MARGINAL UTILITY

• U(x1,x2)
• MU1?U/?x1 ? ?UMU1.?x1
• MU2?U/?x2 ? ?UMU2.?x2
• Along a particular indifference curve
• ?U 0 MU1(?x1) MU2(?x2)
• ? ?x2/?x1 MRS (-)MU1/MU2

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MARGINAL UTILITY

• E.g. if U(x1,x2) x11/2 x22 then

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MARGINAL UTILITY

• E.g. if U(x1,x2) x11/2 x22 then

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MARGINAL UTILITY

• E.g. if U(x1,x2) x11/2 x22 then

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MARGINAL UTILITY

• E.g. if U(x1,x2) x11/2 x22 then

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MARGINAL UTILITY

• So, if U(x1,x2) x11/2 x22 then

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MARGINAL UTLITIES AND MARGINAL RATE OF
SUBISITUTION

• The general equation for an indifference curve
  is U(x1,x2) º k, a constant
• Totally differentiating this identity gives

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MARGINAL UTLITIES AND MARGINAL RATE OF
SUBISITUTION
rearranging
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MARGINAL UTLITIES AND MARGINAL RATE OF
SUBISITUTION
rearranging
This is the MRS.
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MUs and MRS An example

• Suppose U(x1,x2) x1x2. Then

so
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MUs and MRS An example
U(x1,x2) x1x2
x2
8
MRS(1,8) - 8/1 -8 MRS(6,6) - 6/6
-1.
6
U 36
U 8
x1
1
6
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MONOTONIC TRANSFORMATIONS AND MRS

• Applying a monotonic transformation to a utility
  function representing a preference relation
  simply creates another utility function
  representing the same preference relation.
• What happens to marginal rates-of-substitution
  when a monotonic transformation is applied?
  (Hopefully, nothing)

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MONOTONIC TRANSFORMATIONS AND MRS

• For U(x1,x2) x1x2 the MRS (-) x2/x1
• Create V U2 i.e. V(x1,x2) x12x22 What is
  the MRS for V?
• which is the same as the MRS for U.

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MONOTONIC TRANSFORMATIONS AND MRS

• More generally, if V f(U) where f is a strictly
  increasing function, then

So MRS is unchanged by a positivemonotonic
transformation.